Keywords enzyme kinetics; metabolic or signalling pathways; mathematical modelling; protein associations Correspondence M.. To investigdisassoci-ate a role in coordination for FDSs, we h
Trang 1Michel Thellier1,3, Guillaume Legent1, Patrick Amar2,3, Vic Norris1,3and Camille Ripoll1,3
1 Laboratoire ‘Assemblages mole´culaires: mode´lisation et imagerie SIMS’, Faculte´ des Sciences de l’Universite´ de Rouen,
Mont-Saint-Aignan Cedex, France
2 Laboratoire de recherche en informatique, Universite´ de Paris Sud, Orsay Cedex, France
3 Epigenomics Project, Genopole, Evry, France
Numerous studies have shown that proteins involved
in metabolic or signalling pathways are often
distri-buted nonrandomly, as multimolecular assemblies
[1–15] Such assemblies range from quasi-static,
multi-enzyme complexes (such as the fatty acid synthase or
the a-oxo acid dehydrogenase systems [5]) to transient,
dynamic protein associations [2,3,7,15,16] Comparison
of yeast and human multiprotein complexes has shown
that conservation across species extends from single
proteins to protein assemblies [11] Multi-molecular
assemblies may comprise proteins but also nucleic
acids, lipids, small molecules and inorganic ions Such
assemblies may interact with membranes, skeletal
ele-ments and⁄ or cell organelles [3,4,15,17] They have
been termed metabolons, transducons and
repairo-somes in the case of metabolic pathways [3,10,18–23],
signal transduction [24] and DNA repair [12],
respect-ively, or, more generally, hyperstructures [17,25–28]
According to Srere [3], metabolons are enzyme assemblies in which intermediates are channelled from each enzyme to the next without diffusion of these intermediates into the surrounding cytoplasm [2– 7,9,15,23,29–33] Potential advantages of channelling [7,9,15,30,31,34,35] are (i) reduction in the size of the pools of intermediates (a point, however, contested by some authors [36,37]), (ii) protection of unstable or scarce intermediates by maintaining them in a protein-bound state, (iii) avoidance of an ‘underground’ meta-bolism in which intermediates become the substrates of other enzymes [38], and (iv) protection of the cytoplasm from toxic or very reactive intermediates The terms sta-tic and dynamic channelling have been used to describe, respectively, the channelling in a quasipermanent me-tabolon and in a transient association between two enzymes occurring while the intermediate metabolite is transferred from the first enzyme to the second [39,40]
Keywords
enzyme kinetics; metabolic or signalling
pathways; mathematical modelling; protein
associations
Correspondence
M Thellier, Laboratoire Assemblages
mole´culaires: mode´lisation et imagerie SIMS
FRE CNRS 2829, Faculte´ des Sciences de
l’Universite´ de Rouen, F-76821
Mont-Saint-Aignan Cedex, France
Fax: +33 2 35 14 70 20
Tel: +33 2 35 14 66 82
E-mail: Michel.Thellier@univ-rouen.fr
(Received 12 January 2006, revised 26 June
2006, accepted 20 July 2006)
doi:10.1111/j.1742-4658.2006.05425.x
A fundamental problem in biochemistry is that of the nature of the coordination between and within metabolic and signalling pathways It is conceivable that this coordination might be assured by what we term func-tioning-dependent structures (FDSs), namely those assemblies of proteins that associate with one another when performing tasks and that disassoci-ate when no longer performing them To investigdisassoci-ate a role in coordination for FDSs, we have studied numerically the steady-state kinetics of a model system of two sequential monomeric enzymes, E1and E2 Our calculations show that such FDSs can display kinetic properties that the individual enzymes cannot These include the full range of basic input⁄ output charac-teristics found in electronic circuits such as linearity, invariance, pulsing and switching Hence, FDSs can generate kinetics that might regulate and coordinate metabolism and signalling Finally, we suggest that the occur-rence of terms representative of the assembly and disassembly of FDSs in the classical expression of the density of entropy production are character-istic of living systems
Abbreviation
FDS, functioning-dependent structure.
Trang 2We propose here to generalize the concept of dynamic
channelling or, more precisely, the concept of a
struc-ture that dynamically and transiently forms to carry out
a process, into that of functioning-dependent structure
(FDS) [41] In other words, an FDS is a dynamic,
multi-molecular structure that assembles when functioning
and that disassembles when no longer functioning, and
thus is created and maintained by the very fact that it is
in the process of accomplishing a task The lifetime of
such a structure may be short or long, depending only
on the duration of the process that is catalysed by the
FDS An FDS catalyses efficiently the processes that
have allowed this FDS to form It can therefore be
viewed as a self-organized structure
Published examples of transient, dynamic
multi-molecular assemblies, that only form in an
activity-dependent manner include: the role of the bifunctional
protein complex cysteine synthetase in the synthesis of
cysteine in Salmonella typhimurium [42]; the
metabo-lite-modulated formation of complexes (especially
binary complexes) of sequential glycolytic enzymes
[4,43,44]; the functional coupling of pyruvate kinase
and creatine kinase via an enzyme–product–enzyme
complex in muscle [45]; the interaction between serine
acetyl-transferase and O-acetylserine(thiol)-lyase in
higher plants [46,47]; the ATP- and pH-dependent
association⁄ dissociation of the V1 and V0 domains of
the yeast vacuolar H+- ATPases [48–50]; the
promo-tion by substrate binding of the assembly of the three
components of protein-mediated exporters involved in
protein secretion in Gram-negative bacteria [51]; the
first step of glycogenolysis in vertebrate muscle tissues
by the sequential formation of a
phosphorylase–glyco-gen complex followed by the binding of phosphorylase
kinase to this previously formed complex [18]; the
clus-tering of the anchoring protein gephyrin with glycine
receptors following glycine receptor activation in
postsynaptic regions of spinal neurons [52–55]; the
clustering of antigen receptors followed by binding of
intracellular proteins, such as protein tyrosine kinases,
to the cytoplasmic portion of the receptors in the case
of signalling through lymphocyte receptors (reviewed
in [56]); the organization of functional rafts in the
plasma membrane upon T-cell activation [57]; the
gly-cine decarboxylase complex in higher plants [58]; the
assembly of water-soluble, cytosolic proteins with the
membrane-anchored flavocytochrome b558for the
cata-lysis of the NADPH-dependent reduction of O2 into
the superoxide anion O2 in stimulated phagocytic cells
[59]; the dynamic association of HSP90 with the
RPM1 disease resistance protein in the response of
Arabidopsis plants to infection by Pseudomonas
syrin-gae [60]; the association of protein complexes with
assembling actin molecules in the lamellipodium tip of moving cells [61]; the clustering of glutamate receptors opposite the largest and most physiologically active sites of presynaptic release [62]; the differential nucleo-tide-dependent binding of Bfp proteins in the transduc-tion of mechanical energy to the biogenesis machine of Escherichia coli [63] Even the Golgi apparatus of Sac-charomyces cerevisiae can be viewed as a dynamic structure with a size that depends on its functioning such that it grows when it is secreting and shrinks when it is not [64–67]
It is striking that these cellular systems that have very different structures and functions nevertheless exhibit the common behaviour of assembling into tran-sient complexes or FDSs when functioning Why? A fundamental problem in biochemistry is that of coordi-nation The functioning of a protein in a metabolic or signalling pathway in vivo is coordinated with that of the other proteins in the same pathway, and the func-tioning of the pathway itself is coordinated with that
of the other pathways within the cell In metabolic pathways, the regulation needed for such coordination comes in part from the sigmoidal kinetics provided by allosteric enzymes, due to the fact that subunit–subunit interactions are added to the classical enzyme–sub-strate interactions [68] It is therefore tempting to spe-culate that FDSs are involved in the coordination within and between metabolism and signalling
If FDSs are to have a central role in coordination, they should be predicted to generate regulatory kinet-ics via the enzyme–enzyme interactions that constitute them In the following, we have endeavoured to test this prediction by numerically studying the steady-state kinetics of a model system of two sequential
monomer-ic enzymes, E1 and E2, which, when free, are of the Michaelis–Menten type (i.e., with a single substrate-binding site and no regulatory site) Our results show that the metabolite-induced association of these two enzymes into an FDS [20] may, under steady-state con-ditions, confer to the FDS basic regulatory kinetic fea-tures, that the individual enzymes lack These include the full range of input⁄ output characteristics found in electronic circuits such as a linear relationship between input and output, an output limited to a narrow range
of inputs, a constant output whatever the input, and even switch-like behaviours (Fig 1) Hence a metabo-lite-induced FDS could generate a wide variety of kin-etics that could serve as signals
Modelling a two-enzyme FDS The different substances and reactions that can possibly take place when an FDS is involved in the overall
Trang 3trans-formation of an initial substrate, S1, into a final
prod-uct, S3, via reactions catalysed by two enzymes, E1and
E2, are represented in Fig 2 In total, 29 reactions act
on 17 substances (free substances and complexes) and,
to account for a formation of the FDS solely dependent
on its activity, the reaction E1+E2¼ E1E2 does not
exist in this scheme Note that the symbols used in
Fig 2 to describe the complexes are such that E1S2E2
and E1E2S2 mean that S2 is bound to the catalytic site
of E1 or of E2, respectively, within the FDS, etc To
write down the steady-state conditions of functioning of
the system (further details given in Appendix), (i) we
assume that external mechanisms supply S1and remove
S3 as and when they are consumed and produced,
respectively, such that S1 is maintained at a constant
concentration and S3 at a zero concentration, and (ii)
we use the set of algebraic equations obtained by
wri-ting down the mass balance of the 15 other species
involved For convenience, we have reasoned using
di-mensionless variables (note that capital letters are used
for chemical species and small letters for dimensionless
concentrations) We have also taken into account the
fact that the law of mass action has to be satisfied
what-ever the pathway from S1 to S3 When all calculations
are carried out for any given value of the concentration,
s1, of S1, the steady-state rate of transformation of S1 into S3 is calculated as corresponding to both the rate
of consumption of S1, v(s1), and the rate of production
of S3, v(s3), and the shape of the curves {s1, v(s1)} is examined in cases involving either free enzymes alone
or an FDS with free enzymes
It is worth noting that it would only be necessary to add a few more reactions to Fig 2 to describe the interaction of these enzymes with other proteins or molecules and hence study systems in which, for exam-ple, small proteins contribute to the formation of the enzyme–enzyme complexes [15]; the theoretical treat-ment would be longer but otherwise essentially the same as that followed here
Results
Kinetics of the overall reaction of transformation
of S1into S3 The system with only the free enzymes, E1and E2 The overall rate of functioning of two free sequential enzymes of the Michaelis–Menten type involved in a metabolic pathway has already been computed as a function of the concentration of initial substrate under
output
input
output
input output
input
output
input
D C
(b)
(a)
Fig 1 Classical input ⁄ output relationships in electrical circuits (A) Linear response: this behaviour is obtained when a generator is connec-ted to a load (resistor) (B) Constant response: this behaviour is obtained when a source of current is connecconnec-ted to a load; whatever the value of the load, and therefore whatever the value of the potential difference, the current is unchanged (C) Impulse response: the output
is non-null only for a particular value (or a narrow range of values) of the input (D) curve (a): Step response: this behaviour corresponds to a switch from low or null current to high current when the potential difference exceeds a threshold; curve (b): Inverse step response: this behaviour corresponds to a switch from high current to low or null current when the potential difference exceeds a threshold.
Trang 4steady-state conditions [69] The results are summarized
in Fig 3A Briefly, curves monotonically increasing up
to a plateau and exhibiting no inflexion points were
obtained for all parameter values tested Occasionally,
the shape of these curves was close to that of a
hyper-bola Cases existed (with the smallest K2 values in
Fig 3A) in which the overall rate of reaction became a
quasi-linear function of the concentration of initial
sub-strate, s1, almost up to the plateau (which never occurs
when a single enzyme is involved) Hence, under certain
conditions, free enzymes can generate signals or other
behaviours corresponding to a linear relationship
between input (concentration of first substrate) and
output (rate of production of final product) (Fig 1A)
The system with an FDS
At some parameter values, in the case of an FDS, the
{s1, v(s1)} curves were similar to those obtained with
the free enzymes, i.e., they increased monotonically
without an inflexion point up to a plateau and
some-times exhibited an extended linear response with v(s1)
proportional to s1 over a large range of s1 values
(Fig 3B, curves c and d) However, at other parameter
values, the {s1, v(s1)} curves exhibited a variety of
forms that were not found with the free enzymes For
instance, in Fig 3B, the curves (a) and (b) exhibited
substrate-inhibition behaviour, i.e., with increasing s1,
the rate of consumption of S1 initially increased then,
after reaching a maximal value, decreased
The occurrence of {s1, v(s1)} curves with a
substrate-inhibition shape was examined further (Fig 4) At
some parameter values, with increasing s1, the rate of
consumption of S1 decreased to almost zero (Fig 4A) This means that this FDS system exhibited a sort of inversed behaviour in which it was active at low s1 val-ues (except at the very lowest s1values) and inactive at the high s1 values This corresponds to the scenario in Fig 1C in which an increasing input leads to an output in the form of a spike or impulse Another case
in which an increasing input leads to an output in the form of an impulse (i.e., corresponding to the scenario
in Fig 1C) is depicted in Fig 4B
At other values of the parameters, with increasing
s1, the rate of consumption of S1 again increased, reached a maximal value, then decreased, whilst at sat-urating values of s1 the rate of consumption of S1 reached a plateau (instead of decreasing to zero) (Fig 4C) Moreover, at the largest K1 values (K1¼
104), the rate of consumption of S1almost immediately reached the plateau (Fig 4C, curve d), which means that the response of the system became effectively independent of s1 (except again at the very lowest s1 values) This corresponds to the scenario in Fig 1B in which the output is independent of the input
A curve is shown (Fig 4D) that over a wide range
of low values of s1 has a relatively constant and high rate of consumption of S1 but that with higher values
of s1 drops rapidly to a constant and low rate of con-sumption This resembles the switch shown in Fig 1D curve (b)
Curves with a sigmoid shape, i.e., resembling the switch shown in Fig 1D curve (a), were sometimes obtained (Fig 5A) At the parameter values tested, however, the adjustment of the curve to a Hill function v(s1)¼ vmaxÆ(s1)n⁄ [(k)n+(s1)n] (in which n is the Hill
Fig 2 The scheme of the reactions involved in the functioning of our model of a two-enzyme FDS The system comprises 17 different chemical species (free enzymes, free substrates or products, and binary, ternary or quaternary complexes) indicated
in the green circles These species are linked to one another by 29 chemical reac-tions numbered R1to R29as indicated in the rectangles.
Trang 5coefficient, vmax is the maximal rate of reaction and k
is the value of s1 that gives v(s1)¼ 0.5Ævmax) was not
entirely satisfactory because a perfect straight line was
not obtained (r2¼ 0.985) when using the Hill system
of coordinates, {log s1, log [v(s1)⁄ (vmax–v(s1))]}
(Fig 5B); moreover, the sigmoidicity was rather weak
(Hill coefficient equal to only 1.47)
There were cases in which even more complicated
responses occurred For example, in Fig 6 in which
K10 was varied from 1 to 103 and in which all the
other parameters have the values given in the figure
caption, a {s1, v(s1)} curve similar to those in Fig 4C
and with a low plateau value was observed with the
smallest K10 values (Fig 6, curve a) while the
sub-strate-inhibition effect was less and the plateau was
higher with increasing K10 values (Fig 6, curve b)
Finally, with the highest values of K10(Fig 6, curves c
and d), the {s1, v(s1)} curves increased monotonically
to a plateau but with two inflexion points that
con-ferred on them a dual-phasic aspect Dual-phasic
kin-etic curves are often exhibited by both natural and
artificial enzymatic and transport systems [70–72];
although the functional advantage of such kinetics is
not clear, it is interesting that this complex behaviour
can be revealed by an FDS with as few as two
enzymes
Discussion
The consequences of channelling on metabolism have
been extensively explored by modelling In channelling,
the intermediate metabolites are confined to very small
volumes within a metabolon and have short half-lives
It may therefore be invalid to assume that the local statistical distribution of any molecule is Poissonian and therefore that the classical macroscopic law of kinetics can be used to describe the reaction rates [29,73–75] Indeed, certain models based on this invalid assumption may even lead to an apparent violation of the second law of thermodynamics [73] The model developed here is based on the classical macroscopic laws of kinetics but, importantly, is self-consistent in the sense that it uses the same assumptions to deter-mine and compare the kinetics of two enzymes freely diffusing or assembled into a FDS
Numerous command or control devices used in engineering are made from elements with input⁄ out-put functions as shown in Fig 1 In electronics, these functions include the linear function obtained when a source of potential difference is connected to
a resistor (Fig 1A), the constant function obtained when a current source is connected to a resistor (Fig 1B), the impulse function (Fig 1C) and the increasing (Fig 1D, curve a) or decreasing (Fig 1D, curve b) step function We have shown here that the assembly of only two enzymes can result in a variety
of input⁄ output relationships including, importantly, those with characteristics similar to these basic func-tions Hence, the assembly of just two enzymes could provide a macromolecular mechanism for control processes This is illustrated by the following exam-ples The substrate concentration could be encoded
in a linear response (Fig 1A) (Note that we occa-sionally obtained linear responses from a system of
0
0.1
0.2
2 0 1
0 0
0 0.06 0.12 0.18
1 0 5
0 0
A
s1
B
a
b
c
d
a
b
c
d
e
s1
Fig 3 Examples of computed {s 1 , v(s 1 )} curves (A) Case of a system made of two free enzymes: the parameter values are e 1t ¼ e 2t ¼ 0.5,
K ¼ 100, k 1r ¼ 1 (Eqn A6), k 2r ¼ 100, k 3r ¼ k 4r ¼ k 9r ¼ k 10r ¼ 1, k 4f calculated according to Eqn (A25), K1¼ 10, K 3 ¼ 100, K 9 ¼ K 10 ¼ 1 and
K2¼ 0.10 (curve a), 0.05 (curve b), 0.01 (curve c), 0.001 (curve d) and 0.0001 (curve e) Modified from [69] (B) Case of a two-enzyme FDS: the parameter values are e 1t ¼ e 2t ¼ 0.5, K ¼ 100, k 1r ¼ 1 (Eqn A6), k 2r ¼ 100, k 3r ¼ k 4r ¼ k 5r ¼ k 6r ¼ k 7r ¼ k 8r ¼ k 9r ¼ k 10r ¼ k 11r ¼
k12r¼ k 13r ¼ k 14r ¼ k 15r ¼ k 16r ¼ k 17r ¼ k 18r ¼ k 19r ¼ k 20r ¼ k 21r ¼ k 22r ¼ k 23r ¼ k 24r ¼ k 25r ¼ k 26r ¼ k 27r ¼ k 28r ¼ k 29r ¼ 1, K1¼ 10, K2¼ 0.01, K5¼ 1000, K 3 ¼ K 10 ¼ K 11 ¼ K 12 ¼ K 13 ¼ K 15 ¼ K 17 ¼ K 29 ¼ 1, K 27 ¼ 100, K 9 ¼ 10 (curve a), 10 2 (curve b), 10 3 (curve c), 10 4 (curve d) and all the other K j calculated as indicated in Eqns (A25) to (A27) and Table A2.
Trang 6two enzymes that diffused freely, i.e., without FDS.)
Homeostasis results when, despite the concentration
of the initial substrate, s1, varying, the rate of
pro-duction of the final product is constant (Fig 1B)
An impulse that could constitute a signal, results
when, at a narrow range of low concentrations of
substrate s1, the rate of production of the final
prod-uct takes the form represented in Fig 1C (Fig 4A,B
show a more realistic representation) A switch as
represented in Fig 1D (curve a) could be based on
the sigmoid curve in the production rate A switch
from a high rate to a low rate of production occurs
when s1 exceeds the threshold s0 at the inflection point (Fig 4D) and this could correspond to a sub-strate-inhibition behaviour Hence the assembly of two enzymes into an FDS could allow a switch behaviour Alternatively, it could allow this enzyme system to be efficient at a low substrate concentra-tion but not at a high concentraconcentra-tion where the sub-strate would become available for enzymes in a different metabolic pathway
A strongly sigmoid curve from low to high rates of production was not revealed by our calculations (see above) Weakly sigmoid curves from low to high rates
0
0.04
0.08
4 0 2
0 0
s1
C
a
b
c
d
s1 0
0.02 0.04
1 0 5
0 0
B
0 0.01 0.02
1 0 5
0 0
s1
D
0.0000
0.0004
0.0008
1 0 5
0 0
A
s1
Fig 4 Various types of substrate-inhibition {s 1 , v(s 1 )} curves computed in the case of a two-enzyme FDS (A) Example of an almost total inhi-bition at high s1values (impulse behaviour): the parameter values are e1t¼ e 2t ¼ 0.5, K ¼ 100, k 1r ¼ 1 (Eqn A6), k 2r ¼ 10 4 , k3r¼ k 4r ¼
k5r¼ k 6r ¼ k 7r ¼ k 8r ¼ k 9r ¼ k 10r ¼ k 11r ¼ k 12r ¼ k 13r ¼ k 14r ¼ k 15r ¼ k 16r ¼ k 17r ¼ k 18r ¼ k 19r ¼ k 20r ¼ k 21r ¼ k 22r ¼ k 23r ¼ k 24r ¼ k 25r ¼ k 26r ¼
k 27r ¼ k 28r ¼ k 29r ¼ 1, K 1 ¼ 10, K 2 ¼ 0.0001, K 5 ¼ 10 6 , K 3 ¼ K 9 ¼ K 10 ¼ K 11 ¼ K 12 ¼ K 13 ¼ K 17 ¼ 1, K 15 ¼ K 27 ¼ 100, K 29 ¼ 1000 and all the other K j calculated as indicated in Eqns (A25) to (A27) and Table A2 (B) Another example of an impulse behaviour: the parameter values are e1t¼ e 2t ¼ 0.5, K ¼ 1000, k 1r ¼ 1 (Eqn A6), k 2r ¼ 10 4 , k3r¼ k 4r ¼ k 5r ¼ k 6r ¼ k 7r ¼ k 8r ¼ k 9r ¼ k 10r ¼ k 11r ¼ 1, k 12r ¼ 10 3 ,
k 13r ¼ k 14r ¼ k 15r ¼ k 16r ¼ k 17r ¼ k 18r ¼ k 19r ¼ k 20r ¼ k 21r ¼ k 22r ¼ k 23r ¼ k 24r ¼ k 25r ¼ k 26r ¼ k 27r ¼ k 28r ¼ 1, k 29r ¼ 10 4 , K 1 ¼ 10, K 2 ¼ 0.0001, K 3 ¼ 1000, K 5 ¼ 10 6
, K 9 ¼ K 10 ¼ K 11 ¼ 1, K 12 ¼ 0.001, K 13 ¼ 100, K 15 ¼ 1000, K 17 ¼ 1, K 27 ¼ 100, K 29 ¼ 10000 and all the other
Kjcalculated as indicated in Eqns (A25) to (A27) and Table A2 (C) Examples of an only partial inhibition at high s1values: the parameter val-ues are e1t¼ e 2t ¼ 0.5, K ¼ 100, k 1r ¼ 1 (Eqn A6), k 2r ¼ 100, k 3r ¼ k 4r ¼ k 5r ¼ k 6r ¼ k 7r ¼ k 8r ¼ k 9r ¼ k 10r ¼ k 11r ¼ k 12r ¼ k 13r ¼ k 14r ¼
k 15r ¼ k 16r ¼ k 17r ¼ k 18r ¼ k 19r ¼ k 20r ¼ k 21r ¼ k 22r ¼ k 23r ¼ k 24r ¼ k 25r ¼ k 26r ¼ k 27r ¼ k 28r ¼ k 29r ¼ 1, K 1 ¼ 10 (curve a), 10 2
(curve b), 103 (curve c) and 10 4 (curve d), K2¼ 0.01, K 5 ¼ 10 3 , K9¼ 10, K 3 ¼ K 10 ¼ K 11 ¼ K 12 ¼ K 13 ¼ K 15 ¼ K 17 ¼ K 29 ¼ 1, K 27 ¼ 100 and all the other
Kjcalculated as indicated in Eqns (A25) to (A27) and Table A2 (D) Example of an inversed step response: the parameter values are e1t¼
e 2t ¼ 0.5, K ¼ 1000, k 1r ¼ 1 (Eqn A6), k 2r ¼ 10 4 , k 3r ¼ k 4r ¼ k 5r ¼ k 6r ¼ k 7r ¼ k 8r ¼ k 9r ¼ k 10r ¼ k 11r ¼ 1, k 12r ¼ 10 3 , k 13r ¼ k 14r ¼ k 15r ¼
k 16r ¼ k 17r ¼ k 18r ¼ k 19r ¼ k 20r ¼ k 21r ¼ k 22r ¼ k 23r ¼ k 24r ¼ k 25r ¼ k 26r ¼ k 27r ¼ k 28r ¼ 1, k 29r ¼ 10 4
, K 1 ¼ 10, K 2 ¼ 0.0001, K 3 ¼ 75, K 5 ¼
10 6 , K9¼ K 10 ¼ K 11 ¼ 1, K 12 ¼ 0.001, K 13 ¼ 100, K 15 ¼ 1000, K 17 ¼ 1, K 27 ¼ 100, K 29 ¼ 10000 and all the other K j calculated as indicated
in Eqns (A25) to (A27) and Table A2.
Trang 7of production were sometimes observed with Hill
coef-ficients of less than 2 (Fig 5) but these could not
con-stitute switches Compared with the sigmoidicity of
allosteric enzymes [68], that of a two-enzyme FDS –
the only type tested here – is poor Experimental
results are consistent with this because the formation
of a protein–protein complex of serine acetyl
trans-ferase with O-acetylserine(thiol)-lyase strongly modifies
the kinetic properties of the first enzyme and results in
a transition from a typical Michaelis–Menten beha-viour to a behabeha-viour displaying positive cooperativity with respect to serine and acetyl-CoA with a Hill coef-ficient in the range of 1.3–2.0 [47]
It is probable that many more types of FDSs exist than those found so far experimentally (see above) Indeed, many FDSs may have escaped detection pre-cisely because they tend to dissociate as the substrate concentration decreases, as generally occurs during
in vitro studies It may even turn out that most enzymes and other proteins such as those involved in signalling assemble into FDSs in vivo when function-ing These FDSs may be connected to more permanent structures such as membranes and the cytoskeleton They may even be connected to one another to form a network integrating FDSs responsible for metabolism and for signal transduction [11,76] Such a vision of intracellular organization is supported by many studies showing the recruitment of proteins into functional structures (reviewed in [3–5]) and the coordination of multiple functions via the formation of networks of signalling complexes [11,16,77–79] More than 50 dif-ferent types of protein assemblies, containing up to 35 proteins, have been identified in functions that include transcription regulation, cell-cycle⁄ cell-fate control, RNA processing, and protein transport [13] It could
be argued that the concept of FDS should not be lim-ited to the intracellular level Indeed, a concept similar
to that of the FDS has been employed at the multi-cellular level to explain how neurones participate in
0
0.04
0.08
2 0 1
0 0
-3 -2 -1 0 1 2
B
log s1
s 1
v(s1)
Fig 5 Example of a sigmoid {s1, v(s1)} curve computed in the case of a two-enzyme FDS The parameter values are e1t¼ e 2t ¼ 0.5,
K ¼ 100, k 1r ¼ 1 (Eqn A6), k 2r ¼ 10, k 3r ¼ k 4r ¼ k 5r ¼ k 6r ¼ k 7r ¼ k 8r ¼ k 9r ¼ k 10r ¼ k 11r ¼ k 12r ¼ k 13r ¼ k 14r ¼ k 15r ¼ k 16r ¼ k 17r ¼ k 18r ¼
k 19r ¼ k 20r ¼ k 21r ¼ k 22r ¼ k 23r ¼ k 24r ¼ k 25r ¼ k 26r ¼ k 27r ¼ k 28r ¼ k 29r ¼ 1, K 1 ¼ K 2 ¼ 0.1, K 3 ¼ 10, K 5 ¼ 1000, K 9 ¼ K 10 ¼ K 11 ¼ K 12 ¼
K13¼ K 15 ¼ K 17 ¼ K 29 ¼ 1, K 27 ¼ 100 and all the other K j calculated as indicated in Eqns (A25) to (A27) and Table A2 (A) Curve represented using the direct system of coordinates, {s 1 , v(s 1 )} (B) Curve represented using the Hill system of coordinates, {log s 1 , log z 1 } with
z 1 ¼ {v(s 1 ) ⁄ [v max –v(s 1 )]}; from the slope of the dashed regression line fitted to the curve, the Hill coefficient was estimated to be of the order
of 1.47.
0
0.2
0.4
2 0 1
0 0
a
b
c
d
v(s1)
s1
Fig 6 Examples of dual-phasic {s1, v(s1)} curves computed in the
case of a two-enzyme FDS The parameter values are e 1t ¼ e 2t ¼
0.5, K ¼ 100, k 1r ¼ 1 (Eqn A6), k 2r ¼ 100, k 3r ¼ k 4r ¼ k 5r ¼ k 6r ¼
k7r¼ k 8r ¼ k 9r ¼ k 10r ¼ k 11r ¼ k 12r ¼ k 13r ¼ k 14r ¼ k 15r ¼ k 16r ¼ k 17r ¼
k 18r ¼ k 19r ¼ k 20r ¼ k 21r ¼ k 22r ¼ k 23r ¼ k 24r ¼ k 25r ¼ k 26r ¼ k 27r ¼
k28r¼ k 29r ¼ 1, K 1 ¼ K 9 ¼ 10, K 2 ¼ 0.01, K 3 ¼ K 11 ¼ K 12 ¼ K 13 ¼
K15¼ K 17 ¼ K 29 ¼ 1, K 5 ¼ 1000, K 27 ¼ 100, K 10 ¼ 1 (curve a), 10
(curve b), 10 2 (curve c) and 10 3 (curve d) and all the other K j
calcu-lated as indicated in Eqns (A25) to (A27) and Table A2.
Trang 8different assemblies at different times depending on the
task to be carried out [80]
Biochemists are familiar with the Structurefi
Func-tion relaFunc-tionship with respect to proteins or other
active molecules or cell substructures They are less
familiar with the idea that the very functioning of
these cellular components may result in their
assem-bling into a dynamic structure from which a better or
even a new functioning emerges In this case, the
rela-tionship above must be changed into
This leads to the intuition that the very existence of
such a self-organizing relationship in a system is an
indication that this system is a living one To try to
express this quantitatively, consider the density of
entropy production in a process involving an FDS
According to the second law of thermodynamics, the
functioning of any system entails a positive production
of entropy that can be written as a bilinear form of the
flux densities of the processes and their conjugated
driving forces [81] Whichever reaction pathway in our
system is chosen to connect S1 and S3 (Fig 2), under
steady-state conditions the only molecules that undergo
transformation are S1and S3while the other molecules
remain unchanged Hence, the corresponding density of
entropy production, r, is that of the overall reaction of
transformation of S1 into S3, and r does not depend
on whether the system is catalysed via free enzymes or
an FDS Out of steady state, however, the situation is
different because the free enzymes, E1 and E2, can act
immediately on their substrates whereas the FDS
enzymes must assemble into an FDS before they can
act Consequently, if r is expressed in the standard
way, terms representing the entropic cost of FDS
assembly⁄ disassembly are present only in the
descrip-tion of living systems
Acknowledgements
We thank Jacques Ricard and Derek Raine for helpful
comments and criticisms
References
1 Mowbray J & Moses V (1976) The tentative
identifica-tion in Escherichia coli of a multi-enzyme complex with
glycolytic activity Eur J Biochem 66, 25–36
2 Srivastava DK & Bernhard SA (1986) Metabolic transfer
via enzyme-enzyme complexes Science 234, 1081–1086
3 Srere PA (1987) Complexes of sequential metabolic enzymes Annu Rev Biochem 56, 21–56
4 Keleti T, Ova´di J & Batke J (1989) Kinetic and physico-chemical analysis of enzyme complexes and their possi-ble role in the control of metabolism Prog Biophys Mol Biol 53, 105–152
5 Ova´di J, Tompa P, Ve´rtessy B, Orosz F, Keleti T & Welch GR (1989) Transient time analysis of substrate channelling in interacting enzyme systems Biochem J
257, 187–190
6 Srere PA & Ova´di J (1990) Enzyme–enzyme interactions and their metabolic role FEBS Lett 268, 360–364
7 Ova´di J (1991) Physiological significance of metabolic channelling J Theor Biol 152, 1–22
8 Hrazdina G & Jensen RA (1992) Spatial organization
of enzymes in plant metabolic pathways Annu Rev Plant Physiol Mol Biol 43, 241–267
9 Mathews CK (1993) The cell – bag of enzymes or net-work of channels? J Bacteriol 175, 6377–6381
10 Mitchell CG (1996) Identification of a multienzyme complex of the tricarboxylic acid cycle enzymes contain-ing citrate synthase isoenzymes from Pseudomonas aeruginosa Biochem J 313, 769–774
11 Gavin AC, Bo¨sche M, Krause R, Grandi P, Marzloch
M, Bauer A, Schultz J, Rick JM, Michon AM, Cruciat
CM et al (2002) Functional organization of the yeast proteome by systematic analysis of protein complexes Nature 415, 141–147
12 Minsky A, Shimoni E & Frenkiel-Krispin D (2002) Stress, order and survival Nature Rev Mol Cell Biol 3, 50–60
13 Spirin V & Mirny LA (2003) Protein complexes and functional modules in molecular networks Proc Natl Acad Sci USA 100, 12123–12128
14 Champion MM, Campbell CS, Siegele DA, Russell
DH & Hu JC (2003) Proteome analysis of Escherichia coli K-12 by two-dimensional native-state chromatography and MALDI-MS Mol Microbiol 47, 383–396
15 Winkel BSJ (2004) Metabolic channelling in plants Annu Rev Plant Biol 55, 85–107
16 Aloy P, Bo¨ttcher B, Ceulemans H, Leutwein C, Mellwig
C, Fischer S, Gavin AC, Bork P, Superti-Furga G, Serr-ano L & Russell RB (2004) Structure-based assembly of protein complexes in yeast Science 303, 2026–2029
17 Norris V & Fishov I (2001) Membrane domains, hyper-structures and cell division Biochimie 83, 91–98
18 Shmelev VK & Serebrenikova TP (1997) A study of supramolecular organization of glycogenolytic enzymes
in vertebrate muscle tissue Biochem Mol Biol Int 43, 867–872
19 Velot C, Mixon MB, Teige M & Srere PA (1997) Model
of a quinary structure between Krebs TCA cycle enzymes: a model for the metabolon Biochemistry 36, 14271–14276
Function Structure
Trang 920 Norris V, Gascuel P, Guespin-Michel J, Ripoll C &
Saier MH Jr (1999) Metabolite-induced metabolons: the
activation of transporter-enzyme complexes by substrate
binding Mol Microbiol 31, 1592–1595
21 Pauwels K, Abadjieva A, Hilven P, Stankiewicz A &
Crabeel M (2003) The N-Acetylglutamate synthase⁄
N-Acetylglutamate kinase metabolon of Saccharomyces
cerevisiaeallows coordinated feedback regulation of the
first two steps in arginine biosynthesis Eur J Biochem
270, 1014–1024
22 Alvarez BV, Vilas GL & Casey JR (2005) Metabolon
disruption: a mechanism that regulates bicarbonate
transport EMBO J 24, 2499–2511
23 Jørgensen K, Rasmussen AV, Morant M, Nielsen AH,
Bjarnholt N, Zagrobelny M, Bak S & Møller BL (2005)
Metabolon formation and metabolic channeling in the
biosynthesis of plant natural products Curr Opin Plant
Biol 8, 280–291
24 Trewavas AJ & Malho´ R (1997) Signal perception and
transduction: the origin of the phenotype Plant Cell 9,
1181–1195
25 Le Sceller L, Ripoll C, Demarty M, Cabin-Flaman A,
Nystrom A, Saier MH Jr & Norris V (2000) Modelling
bacterial hyperstructures with cellular automata
Interjournal of Complex SystemsArticle 366 http://
www.interjournal.org
26 Norris V, Amar P, Ballet P, Bernot G, Delaplace F,
Demarty M, Giavitto JL, Ripoll C, Thellier M &
Zemirline A (2002) Hyperstructures In Proceedings of
the Autrans Seminar on Modelling and Simulation of
Bio-logical Processes in the Context of Genomics(Amar P,
Ke´pe`s F, Norris V & Tracqui P, eds), pp 169–191
Barne´oud, Bonchamp-le`s-Laval, France
27 Guzma´n EC, Guarino E, Riola J, Caballero JL &
Jime´-nez-Sa´nchez A (2003) Ribonucleoside diphosphate
reductase is a functional and structural component of
the replication hyperstructure in Escherichia coli Recent
Res Devel Mol Biol 1, 29–43
28 Thellier M (2003) From a static to a dynamic
descrip-tion of living systems Nova Acta Leopoldina NF 88
332, 11–15
29 Westerhoff HV & Welch GR (1992) Enzyme
organi-zation and the direction of metabolic flow:
physicochemi-cal considerations Curr Top Cell Regul 33, 361–390
30 Mendes P, Kell DB & Welch GR (1995) Metabolic
channelling in organized enzyme systems: experiments
and models In Enzymology in vivo (Advances in
Mole-cular and Cellular Biology series) (Brindle K, ed.), pp
1–19 JAI Press, London
31 Anderson KS (1999) Fundamental mechanisms of
sub-strate channelling Methods Enzymol 308, 111–145
32 James CL & Viola RE (2002) Production and
charac-terization of bifunctional enzymes Substrate
channel-ling in the aspartate pathway Biochemistry 41, 3726–
3731
33 Maher AD, Kuchel PW, Ortega F, de Atauri P, Centelles J & Cascante M (2003) Mathematical modelling of the urea cycle A numerical investigation into substrate channelling Eur J Biochem 270, 3953– 3961
34 Mendes P, Kell DB & Westerhoff HV (1992) Channel-ling can decrease pool size Eur J Biochem 204, 257– 266
35 Ve´rtessy BG & Vas M (1992) Metabolite channeling versus free diffusion: reinterpretation of aldolase-catalysed inactivation of glyceraldehydes-3-phosphate dehydrogenase Biochem J 286, 977–980
36 Cornish-Bowden A (1991) Failure of channelling to maintain low concentrations of metabolic intermediates Eur J Biochem 195, 103–108
37 Cornish-Bowden A & Cardenas ML (1993) Channelling can affect concentrations of metabolic intermediates at constant net flux: artefact or reality? Eur J Biochem 213, 87–92
38 D’Ari R & Casadesus J (1998) Underground metabo-lism Bioessays 20, 181–186
39 Friedrich P (1974) Dynamic compartmentation in sol-uble enzyme systems Acta Biochim Biophys Acad Sci Hung 9, 159–173
40 Friedrich P (1984) Supramolecular Enzyme Organization: Quaternary Structure and Beyond Pergamon Press⁄ Akade´miai Kiado´, Oxford⁄ Budapest
41 Thellier M, Legent G, Norris V, Baron C & Ripoll C (2004) Introduction to the concept of ‘functioning-dependent structures’ in living cells CR Biologies 327, 1017–1024
42 Kredich NM, Becker MA & Tomkins GM (1969) Purifi-cation and characterization of cysteine synthetase, a bifunctional protein complex, from Salmonella typhimur-ium J Biol Chem 244, 2428–2439
43 Ova´di J (1988) Old pathway-new concept: control
of glycolysis by metabolite–modulated dynamic enzyme associations Trends Biochem Sci 13, 486– 490
44 Torshin I (1999) Activating oligomerization as interme-diate level of signal transduction: analysis of protein-protein contacts and active sites in several glycolytic enzymes Front Biosci 4D, 557–570
45 Dillon PF & Clark JF (1990) The theory of diazymes and functional coupling of pyruvate kinase and creatine kinase J Theor Biol 143, 275–284
46 Droux M, Martin J, Sajus P & Douce R (1992) Purifica-tion and characterizaPurifica-tion of O-acetylserine (thiol) lyase from spinach chloroplasts Arch Biochem Biophys 295, 379–390
47 Droux M, Ruffet ML, Douce R & Job D (1998) Inter-action between serine acetyl-transferase and O-acetylser-ine (thiol) lyase in higher plants Structure and kO-acetylser-inetic properties of the free and bound enzymes Eur J Bio-chem 255, 235–245
Trang 1048 Kane PM (1995) Disassembly and reassembly of the
yeast vacuolar H+-ATPase in vivo J Biol Chem 270,
17025–17032
49 Parra KJ & Kane PM (1998) Reversible association
between the V1 and V0 domains of yeast vacuolar H+
-ATPase is an unconventional glucose-induced effect
Mol Cell Biol 18, 7064–7074
50 Kane PM & Parra KJ (2000) Assembly and regulation of
the yeast vacuolar H+-ATPase J Exp Biol 203, 81–87
51 Le´toffe´ S, Delepelaire P & Wandersman C (1996)
Pro-tein secretion in Gram-negative bacteria: assembly of
the three components of ABC protein mediated
expor-ters is ordered and promoted by substrate binding
EMBO J 15, 5804–5811
52 Kirsch J & Betz H (1998) Glycine-receptor activation is
required for receptor clustering in spinal neurones
Nature 392, 717–720
53 Betz H, Kuhse J, Schmieden V, Laube B, Kirsch J &
Harvey RJ (1999) The strychnine-sensitive glycine
recep-tor (GlyR) is a pentameric chloride Ann N Y Acad Sci
868, 667–676
54 Sabatini DK, Barrow RK, Blackshaw S, Burnett PE,
Lai MM, Field ME, Bahr BA, Kirsch J, Betz H &
Snyder SH (1999) Interaction of RAFT1 with Gephyrin
is required for rapamycine-sensitive signalling Science
284, 1161–1164
55 Kins S, Betz H & Kirsch J (2000) Collybistin, a newly
identified brain-specific GEF, induces submembrane
clustering of gephyrin Nat Neurosci 3, 22–29
56 Janeway CA, Travers P, Walport M & Capra JD (1999)
Signalling through lymphocyte receptors In
Immuno-biology, 4th edn, pp 163–193 Elsevier Science Ltd,
London and Garland Pu, New York
57 Tuosto L, Parolini I, Schroder S, Sargiacomo M,
Lanzavecchia A & Viola A (2001) Organization of
plasma membrane functional rafts upon T-cell
activation Eur J Immunol 31, 345–349
58 Douce R, Bourguignon J, Neuburger M & Re´beille´ F
(2001) The glycine decarboxylase system: a fascinating
complex Trends Plant Sci 6, 167–176
59 Vignais PV (2002) The superoxide-generating NADPH
oxidase: structural aspects and activation mechanisms
Cell Mol Life Sci 59, 1428–1459
60 Hubert DA, Tornero P, Belkhadir Y, Krishna P,
Takahashi A, Shirasu K & Dangl JL (2003) Cytosolic
HSP90 associates with and modulates the Arabidopsis
RPM1 disease resistance protein EMBO J 22, 5679–
5689
61 Giannone G, Dubin-Thaler BJ, Do¨bereiner HG, Kieffer
N, Bresnick AR & Sheetz MP (2004) Periodic
lamellipo-dial contractions correlate with rearward actin waves
Cell 116, 431–443
62 Marrus SB & Di Antonio A (2004) Preferential
localiza-tion of glutamate receptors opposite sites of high
presy-naptic release Curr Biol 14, 924–931
63 Crowther LJ, Yamagata A, Craig L, Tainer JA & Donnenberg MS (2005) The ATPase activity of BfpD
is greatly enhanced by zinc and allosteric interactions with other Bfp proteins J Biol Chem 280, 24839– 24848
64 Morin-Ganet MN, Rambourg A, Deitz SB, Franzusoff
A & Ke´pe`s F (2000) Morphogenesis and dynamics of the yeast Golgi apparatus Traffic 1, 56–68
65 Ke´pe`s F (2002) Secretory compartments as instances of dynamic self-evolving structures Acta Biotheor 50, 209– 221
66 Presley JF, Ward TH, Pfeifer AC, Siggia ED, Phair RD
& Lippincott-Schwartz J (2002) Dissection of COPI and Arf1 dynamics in vivo and role in Golgi membrane transport Nature 417, 187–193
67 Rambourg A, Delosme JM, Incitti R, Satiat-Jeunemaıˆ-tre B, Tracqui P & Ke´pe`s F (2002) Modeling the dynamics of secretory compartments In Proceedings of the Autrans Seminar on Modelling and Simulation of Bio-logical Processes in the Context of Genomics(Amar P, Ke´pe`s F, Norris V & Tracqui P, eds), pp 147–168 Barne´oud, Bonchamp-le`s-Laval, France
68 Monod J, Wyman J & Changeux JP (1965) On the nat-ure of allosteric transitions: a plausible model J Mol Biol 12, 88–118
69 Legent G, Thellier M, Norris V & Ripoll C (2006) Steady-state kinetic behaviour of two- or n-enzyme systems made of free sequential enzymes involved in a metabolic pathway CR Biologies 329, doi: 10.10.16/J.CRVI.2006.02.008
70 Epstein E (1966) Dual pattern of ion absorption by plant cells and by plants Nature 212, 457–474
71 Vincent JC & Thellier M (1983) Theoretical analysis of the significance of whether or not enzyme or transport systems in structured media follow Michaelis-Menten kinetics Biophys J 41, 23–28
72 Thellier M, Vincent JC, Alexandre S, Lassalles JP, Desch-revel B, Norris V & Ripoll C (2003) Biological processes
in organised media CR Biologies 326, 149–159
73 Westerhoff HV & Kamp F (1986) Maxwell’s demons in channelled metabolism: paradoxes and their resolution
In Organization of Cell Metabolism (Welch GR & Clegg
JS, eds), pp 339–356 Plenum Press, London
74 Schnell S & Turner TE (2004) Reaction kinetics in intracellular environments with macromolecular crowd-ing: simulations and rate laws Prog Biophys Mol Biol
85, 235–260
75 Turner TE, Schnell S & Burrage K (2004) Stochastic approaches for modelling in vivo reactions Comput Biol Chem 38, 165–178
76 Ripoll C, Norris V & Thellier M (2004) Ion condensa-tion and signal transduccondensa-tion Bioessays 26, 549–557
77 Jordan JD, Landau EM & Iyengar R (2000) Signaling networks: the origin of cellular multitasking Cell 103, 193–200